0
votes
0answers
16 views

How to translate the polar curves up/down and right/left without referring to Cartesian equations?

If I have a polar equation $r(\theta)$, how can I translate it up/down and right/left? We can do this by converting the equations into Cartesian equations and do the translations we want and then ...
1
vote
0answers
36 views

$r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$

What does the following equation represent? $r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$ My approach: I factorized the equation and it became $(a+r\cos\theta)(a-r)=0$ I feel that ...
0
votes
0answers
58 views

Finding an arc-length between 2 points in 3 dimensions

I know how to find an arc-length between two points with coordinates, say $X=(a,b)$ and $Y=(c,d)$. But how do I find the same thing but for, say $X=(a,b,c)$ and $Y=(d,e,f)$? Thanks!
3
votes
1answer
1k views

How to calculate the area between 2 polar curves: $r=\frac{4}{2}-\sin\theta$ and $r=3\sin\theta$?

How to calculate the area between 2 polar curves: $r=2-\sin\theta$ and $r=3\sin\theta$? I know that one curve is a limaçon and the other is a circle. I have them drawn out as well, my only question ...
4
votes
1answer
817 views

Bézier approximation of archimedes spiral?

As part of an iOS app I’m making, I want to draw a decent approximation of an Archimedes spiral. The drawing library I’m using (CGPath in Quartz 2D, which is C-based) supports arcs as well as cubic ...
0
votes
2answers
210 views

Length of a plane curve in polar coordinate

Consider the plane curve $\gamma$ in polar coordinates: $$ r=r_0+e^{\lambda\theta}, \quad \theta_1 \le \theta \le \theta_2, $$ where $r_0,\lambda,\theta_1>0$. Is it possible to compute explicitly ...
1
vote
1answer
359 views

Find the area of the region determined by two curves

Find the area of the region $R$ given by two curves. So the region $R$ describes the area that is common between the two curves: $$\begin{align*} \text{Function 1: } r&= 2\sin(\theta)\\ ...
5
votes
2answers
1k views

Is $r=2\cos(\theta)$ a one-petal polar function?

I'm currently learning about polar functions and their graphs in precalculus, and one of the questions on my homework is to identify the shape of the function $r=2\cos(\theta)$. We were taught that ...
1
vote
2answers
4k views

Difficult conversion from polar equation to rectangular equation.

How do we convert this into rectangular equation? $r=5\theta$
0
votes
2answers
197 views

How do we get the rectangular form of this?

I know if $\sqrt{x^2+y^2} = x$, then the polar equation of this is $r=cos\theta$ So,how to get the rectangular form of this polar equation, is it complicate: $r=cos(10\theta)$
3
votes
3answers
2k views

Writing a Polar Equation for the Graph of an Implicit Cartesian Equation

If $(x^2+y^2)^3=4x^2y^2,$ then $r=\sin 2\theta$ for some $\theta$. Using $r^2=x^2+y^2, x=r\cos\theta,y=r\sin\theta$, it's easy to get $r^2=\sin^22\theta$. But I don't know what to do next, since ...
0
votes
1answer
633 views

definition of sinusoidal curve

I have question related with these two definition: In geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates $$r^n = a^n \cos(n \theta)$$ where $a$ is ...
2
votes
3answers
4k views

Polar to Parametric Equation?

I'm struggling with this problem, I'm still only on part (a). I tried X=rcos(theta) Y=rsin(theta) but I don't think I'm doing it right. Curve C has polar equation ...
1
vote
2answers
814 views

Express this curve in the rectangular form

Express the curve $r = 9/(4+\sin \theta)$ in rectangular form. And what is the rectangular form? if I get the expression in rectangular form, how am I able to convert it back to polar ...
5
votes
6answers
1k views

Why, conceptually, do limaçons $r=a+b\cos\theta$ have dimples when $|\frac{a}{b}|<2$?

Using calculus, I can justify that limaçons—the polar graphs of $r=a+b\cos\theta$ for various nonzero real values of $a$ and $b$—are dimpled when $|\frac{a}{b}|<2$, but that doesn't seem to yield ...
6
votes
2answers
273 views

Need help with Curves and parameterizations

I'm having some trouble solving a couple of problems: I know this one must be pretty easy but can't find the way to solve it. I need to find the arc length of a curve described by $ r=1- ...